6.2 T = 0, 1 Nuclear Matrix Elements in the Berggren Basis
251
differences between matrix elements that arise from the coupling to the continuum
when one-body states are bound or resonance.
Consequently, it is interesting to discuss the absolute total binding energy of twobody states. For this, one fixes the total binding energy of the ground state of twobody systems at −10, −1, or +1 MeV with respect to the core, as in Sect. 6.2.1. One
can then deduce the influence of continuum coupling on effective nuclear matrix
elements from the energy of calculated two-body states.
The orbital angular momentum of involved partial waves cannot be too large, as
otherwise continuum coupling becomes negligible. Alternatively, the total angular
momentum of partial waves cannot be too small either, as otherwise too few twobody states are present and a global analysis can hardly be made. Thus, one only
considers the p 3/2 , d 5/2 , and f 7/2 partial waves of both protons and neutrons, using,
respectively, the cores of 4 He, 16 O, and 40 Ca in model spaces. Indeed, as we will see
below, effects of the continuum coupling are still visible when = 3, whereas the
number of possible total angular momentum J of the two-body states ranges from
2 to 8, which allows extracting more general conclusions from the obtained results.
Results of calculations are shown in Tables 6.1, 6.2, and 6.3. One can see that
the main effect of continuum coupling is to diminish the excitation energy of the
two-body eigenstates of different J values when one goes from well bound to
unbound states. This means that the overall effects of the residual nucleon–nucleon
interaction become weaker and weaker when eigenstates become less and less
bound. This arises because unbound states, and to a lesser extent weakly bound
states, are more spread in space than well-bound states. Moreover, the differences
between excitation energies of the states with the same total angular momentum J
in well-bound, weakly bound, and unbound states can be as large as a few MeVs,
and are typically a few hundreds of keV. Consequently, as the residual nucleon–
nucleon interaction is mainly acting close to the core, the overall effect of the
residual nucleon interaction decreases when two-body systems become more and
more unbound. One can also notice that the relative spacings between the two-body
states of different J values in a given spectrum also decrease when total binding
energy goes from well bound to unbound. This is also due to the fact that the
residual nucleon–nucleon interaction is less and less binding in this situation, so
that eigenstates, which would be degenerate in the absence of two-body interaction,
become closer to each other.
These effects become less and less important when increases. The differences
between the excitation energies of well-bound, weakly bound, and unbound states
with the same J , which vary from 500 keV to 3 MeV when using p 3/2 partial waves,
can reach up to 2 MeV for d 5/2 partial waves. Even for f 7/2 , these differences of
excitation energies may reach several hundreds of keV. This is a large effect with
direct consequences on spectroscopy of weakly bound or resonance many-body
states. The coupling to continuum, which is prominent in p waves and d waves and
remains non-negligible in higher- waves, should be visible experimentally even in
the medium-heavy nuclei.
A similar effect is seen in the particle-emission widths when = 1, 2, whose
variations are comparable to those of excitation energies (see Tables 6.1, 6.2,
251
differences between matrix elements that arise from the coupling to the continuum
when one-body states are bound or resonance.
Consequently, it is interesting to discuss the absolute total binding energy of twobody states. For this, one fixes the total binding energy of the ground state of twobody systems at −10, −1, or +1 MeV with respect to the core, as in Sect. 6.2.1. One
can then deduce the influence of continuum coupling on effective nuclear matrix
elements from the energy of calculated two-body states.
The orbital angular momentum of involved partial waves cannot be too large, as
otherwise continuum coupling becomes negligible. Alternatively, the total angular
momentum of partial waves cannot be too small either, as otherwise too few twobody states are present and a global analysis can hardly be made. Thus, one only
considers the p 3/2 , d 5/2 , and f 7/2 partial waves of both protons and neutrons, using,
respectively, the cores of 4 He, 16 O, and 40 Ca in model spaces. Indeed, as we will see
below, effects of the continuum coupling are still visible when = 3, whereas the
number of possible total angular momentum J of the two-body states ranges from
2 to 8, which allows extracting more general conclusions from the obtained results.
Results of calculations are shown in Tables 6.1, 6.2, and 6.3. One can see that
the main effect of continuum coupling is to diminish the excitation energy of the
two-body eigenstates of different J values when one goes from well bound to
unbound states. This means that the overall effects of the residual nucleon–nucleon
interaction become weaker and weaker when eigenstates become less and less
bound. This arises because unbound states, and to a lesser extent weakly bound
states, are more spread in space than well-bound states. Moreover, the differences
between excitation energies of the states with the same total angular momentum J
in well-bound, weakly bound, and unbound states can be as large as a few MeVs,
and are typically a few hundreds of keV. Consequently, as the residual nucleon–
nucleon interaction is mainly acting close to the core, the overall effect of the
residual nucleon interaction decreases when two-body systems become more and
more unbound. One can also notice that the relative spacings between the two-body
states of different J values in a given spectrum also decrease when total binding
energy goes from well bound to unbound. This is also due to the fact that the
residual nucleon–nucleon interaction is less and less binding in this situation, so
that eigenstates, which would be degenerate in the absence of two-body interaction,
become closer to each other.
These effects become less and less important when increases. The differences
between the excitation energies of well-bound, weakly bound, and unbound states
with the same J , which vary from 500 keV to 3 MeV when using p 3/2 partial waves,
can reach up to 2 MeV for d 5/2 partial waves. Even for f 7/2 , these differences of
excitation energies may reach several hundreds of keV. This is a large effect with
direct consequences on spectroscopy of weakly bound or resonance many-body
states. The coupling to continuum, which is prominent in p waves and d waves and
remains non-negligible in higher- waves, should be visible experimentally even in
the medium-heavy nuclei.
A similar effect is seen in the particle-emission widths when = 1, 2, whose
variations are comparable to those of excitation energies (see Tables 6.1, 6.2,